Question: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{k^2 + 7k}{k^2 + 9k + 14}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{k^2 + 7k}{k^2 + 9k + 14} = \dfrac{(k)(k + 7)}{(k + 2)(k + 7)} $ Notice that the term $(k + 7)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(k + 7)$ gives: $t = \dfrac{k}{k + 2}$ Since we divided by $(k + 7)$, $k \neq -7$. $t = \dfrac{k}{k + 2}; \space k \neq -7$